Square root--obvious

All you are finding out is that the cutoff money level is proportional to the standard deviation, but in terms of the number of players and not the number of hands.

It should be very very obvious:

The number of hands only establishes the amplitute of the cutoff; the number of players is what sets the standard deviation needed to reach the cutoff, if the tourney is structured on some percentage/percentile advance.

You find the percentage and find how many sd units of positive expectation is needed to place X% moving on. Ie what % area of the tail of normal distribution advances?

Then you examine dollar bankroll predictions, over so many hands, with that level of sd units (sigma) of positive fluctuations. No matter how wildly bets are ranged, the median, over several such tournaments or rounds, will be in square root proportion to the number of players, rather than the number of hands. The average bet size will change, thus the standard deviation per hand will change, but the number of hands will remain constant, thus (QED) the most significant number will be the sqr of the number of players. As players tend to realize that tournament strategy is more short term, and game theory, than long term, they will tend to bet more. But the ingredient that remains the same will always be that results are in proportion to the square root of the number of players.